We are interested in finding a jointly optimal scheduling, routing and power control that achieves max-min fair rate allocation in a multi-hop wireless network. This is a highly complex non-convex optimization problem and it has been previously solved only for small networks. We restrict ourselves to symmetric networks with ring and line topologies, and we numerically solve the problem for a large number of nodes. We model point-to- point links as single user Gaussian channels where nodes cannot send and receive at the same time. This type of channel approximates the performance of CDMA networks and performs better than the equivalent 802.11 network. We show that for smaller transmission powers it is optimal to relay over other nodes whereas for high powers it is optimal to send data directly to a destination. We also show when this transition occurs. We analyze the optimal schedule and find that if a node is active, it should send at the maximum power. When a transmission power is small, the optimal schedule is that every second node is sending, and as the power grows, the distance between active nodes grows. Furthermore, in large networks the distance between nodes sending at the same time is never larger than 4.5 times the size of links used (number of nodes spanned by one transmission link), and it converges to that value for large transmission powers.
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